Tensor Linear Regression: Degeneracy and Solution
Tensor Linear Regression: Degeneracy and Solution
复制标题
张量线性回归：简并性和解
DOI：
10.1109/access.2021.3049494
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发表时间：
2021-01-01
期刊：
影响因子：
3.9
通讯作者：
He，Kejun
中科院分区：
文献类型：
--
作者：
Zhou，Ya;Wong，Raymond K. W.;He，Kejun
张量回归是分析多维数组数据的重要且有用的工具。为了处理高维，通常对（惩罚）损失函数中的系数张量参数施加 CANDECOMP/PARAFAC (CP) 低秩约束。然而，除了众所周知的 CP 参数的不可识别性问题之外，我们证明相应的优化可能没有任何可实现的解决方案，因此当这种情况发生时，系数张量的估计没有明确定义。这与低阶张量逼近问题中称为 CP 简并的现象密切相关。在本文中，我们展示了张量回归问题中 CP 简并性的一些有用结果。为了克服与简并性相关的理论和数值问题，我们提供了一种通用的惩罚策略作为简并性的解决方案。相关结果也解释了为什么某些现有方法比其他方法更稳定。还研究了所得估计的渐近性质。进行数值实验来说明我们的发现。
Tensor regression is an important and useful tool for analyzing multidimensional array data. To deal with high dimensionality, CANDECOMP/PARAFAC (CP) low-rank constraints are often imposed on the coefficient tensor parameter in the (penalized) loss functions. However, besides the well-known non-identifiability issue of the CP parameters, we demonstrate that the corresponding optimization may not have any attainable solutions, and thus the estimation of the coefficient tensor is not well-defined when this happens. This is closely related to a phenomenon, called CP degeneracy, in low-rank tensor approximation problems. In this article, we show some useful results of CP degeneracy in the context of tensor regression problems. To overcome the theoretical and numerical issues associated with the degeneracy, we provide a general penalized strategy as a solution to the degeneracy. The related results also explain why some of the existing methods are more stable than the others. The asymptotic properties of the resulting estimation are also studied. Numerical experiments are conducted to illustrate our findings.